(Dissipation property of
Feller process) Let
be a generator of a Feller process in
,
and
is such that
.
Then

Proof

By
definition
and

Proposition

(Martingale property of
Feller process) Let
be a generator of the Feller process
in
,
,
and
is the
-generated
filtration. Then the
process
is an
-martingale.

Conversely, if
and there exists a
such
that
is an
-martingale
for every
then
and
.

Proof

To prove the direct statement we calculate for
:
By definition of
and Markovian property of
,
thus
It remains to note that according to the proposition
(
Backward Kolmogorov for
Feller
process
)-3,
Hence, we make a change
in the
integral:
and
conclude
To prove the converse statement we rewrite the
condition
as
Hence, we verify the conclusion of the converse statement as
follows
Hence,
and
.